Theorem xkohaus 21266

Description: If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
xkohaus	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ^ko 𝑅) ∈ Haus)

Proof of Theorem xkohaus

Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 20945	. . 3 ⊢ (𝑆 ∈ Haus → 𝑆 ∈ Top)
2		xkotop 21201	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
3	1, 2	sylan2 490	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ^ko 𝑅) ∈ Top)
4		eqid 2610	. . . . . . . 8 ⊢ (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
5	4	xkouni 21212	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ^ko 𝑅))
6	1, 5	sylan2 490	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑅 Cn 𝑆) = ∪ (𝑆 ^ko 𝑅))
7	6	eleq2d 2673	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑓 ∈ (𝑅 Cn 𝑆) ↔ 𝑓 ∈ ∪ (𝑆 ^ko 𝑅)))
8	6	eleq2d 2673	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑔 ∈ (𝑅 Cn 𝑆) ↔ 𝑔 ∈ ∪ (𝑆 ^ko 𝑅)))
9	7, 8	anbi12d 743	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) ↔ (𝑓 ∈ ∪ (𝑆 ^ko 𝑅) ∧ 𝑔 ∈ ∪ (𝑆 ^ko 𝑅))))
10		simprl 790	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑅 Cn 𝑆))
11		eqid 2610	. . . . . . . . . . 11 ⊢ ∪ 𝑅 = ∪ 𝑅
12		eqid 2610	. . . . . . . . . . 11 ⊢ ∪ 𝑆 = ∪ 𝑆
13	11, 12	cnf 20860	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:∪ 𝑅⟶∪ 𝑆)
14	10, 13	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓:∪ 𝑅⟶∪ 𝑆)
15		ffn 5958	. . . . . . . . 9 ⊢ (𝑓:∪ 𝑅⟶∪ 𝑆 → 𝑓 Fn ∪ 𝑅)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 Fn ∪ 𝑅)
17		simprr 792	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
18	11, 12	cnf 20860	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔:∪ 𝑅⟶∪ 𝑆)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔:∪ 𝑅⟶∪ 𝑆)
20		ffn 5958	. . . . . . . . 9 ⊢ (𝑔:∪ 𝑅⟶∪ 𝑆 → 𝑔 Fn ∪ 𝑅)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 Fn ∪ 𝑅)
22		eqfnfv 6219	. . . . . . . 8 ⊢ ((𝑓 Fn ∪ 𝑅 ∧ 𝑔 Fn ∪ 𝑅) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
23	16, 21, 22	syl2anc 691	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
24	23	necon3abid 2818	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 ≠ 𝑔 ↔ ¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
25		rexnal 2978	. . . . . . 7 ⊢ (∃𝑥 ∈ ∪ 𝑅 ¬ (𝑓‘𝑥) = (𝑔‘𝑥) ↔ ¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥))
26		df-ne 2782	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ (𝑔‘𝑥) ↔ ¬ (𝑓‘𝑥) = (𝑔‘𝑥))
27		simpllr 795	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑆 ∈ Haus)
28	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑓:∪ 𝑅⟶∪ 𝑆)
29		simprl 790	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑥 ∈ ∪ 𝑅)
30	28, 29	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑓‘𝑥) ∈ ∪ 𝑆)
31	19	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑔:∪ 𝑅⟶∪ 𝑆)
32	31, 29	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑔‘𝑥) ∈ ∪ 𝑆)
33		simprr 792	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑓‘𝑥) ≠ (𝑔‘𝑥))
34	12	hausnei 20942	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Haus ∧ ((𝑓‘𝑥) ∈ ∪ 𝑆 ∧ (𝑔‘𝑥) ∈ ∪ 𝑆 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))
35	27, 30, 32, 33, 34	syl13anc 1320	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))
36	35	expr 641	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → ((𝑓‘𝑥) ≠ (𝑔‘𝑥) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅)))
37	26, 36	syl5bir 232	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅)))
38		simp-4l 802	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑅 ∈ Top)
39	1	ad4antlr 765	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑆 ∈ Top)
40		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑥 ∈ ∪ 𝑅)
41	40	snssd 4281	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {𝑥} ⊆ ∪ 𝑅)
42	11	toptopon 20548	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
43	38, 42	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
44		restsn2 20785	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑥 ∈ ∪ 𝑅) → (𝑅 ↾t {𝑥}) = 𝒫 {𝑥})
45	43, 40, 44	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑅 ↾t {𝑥}) = 𝒫 {𝑥})
46		snfi 7923	. . . . . . . . . . . . . . 15 ⊢ {𝑥} ∈ Fin
47		discmp 21011	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Comp)
48	46, 47	mpbi 219	. . . . . . . . . . . . . 14 ⊢ 𝒫 {𝑥} ∈ Comp
49	45, 48	syl6eqel 2696	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑅 ↾t {𝑥}) ∈ Comp)
50		simprll 798	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑎 ∈ 𝑆)
51	11, 38, 39, 41, 49, 50	xkoopn 21202	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∈ (𝑆 ^ko 𝑅))
52		simprlr 799	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑏 ∈ 𝑆)
53	11, 38, 39, 41, 49, 52	xkoopn 21202	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∈ (𝑆 ^ko 𝑅))
54	10	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 ∈ (𝑅 Cn 𝑆))
55	16	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 Fn ∪ 𝑅)
56		fnsnfv 6168	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn ∪ 𝑅 ∧ 𝑥 ∈ ∪ 𝑅) → {(𝑓‘𝑥)} = (𝑓 “ {𝑥}))
57	55, 40, 56	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑓‘𝑥)} = (𝑓 “ {𝑥}))
58		simprr1 1102	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑓‘𝑥) ∈ 𝑎)
59	58	snssd 4281	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑓‘𝑥)} ⊆ 𝑎)
60	57, 59	eqsstr3d 3603	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑓 “ {𝑥}) ⊆ 𝑎)
61		imaeq1 5380	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑓 → (ℎ “ {𝑥}) = (𝑓 “ {𝑥}))
62	61	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑓 → ((ℎ “ {𝑥}) ⊆ 𝑎 ↔ (𝑓 “ {𝑥}) ⊆ 𝑎))
63	62	elrab 3331	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑥}) ⊆ 𝑎))
64	54, 60, 63	sylanbrc 695	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎})
65	17	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 ∈ (𝑅 Cn 𝑆))
66	21	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 Fn ∪ 𝑅)
67		fnsnfv 6168	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn ∪ 𝑅 ∧ 𝑥 ∈ ∪ 𝑅) → {(𝑔‘𝑥)} = (𝑔 “ {𝑥}))
68	66, 40, 67	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑔‘𝑥)} = (𝑔 “ {𝑥}))
69		simprr2 1103	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑔‘𝑥) ∈ 𝑏)
70	69	snssd 4281	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑔‘𝑥)} ⊆ 𝑏)
71	68, 70	eqsstr3d 3603	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑔 “ {𝑥}) ⊆ 𝑏)
72		imaeq1 5380	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑔 → (ℎ “ {𝑥}) = (𝑔 “ {𝑥}))
73	72	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑔 → ((ℎ “ {𝑥}) ⊆ 𝑏 ↔ (𝑔 “ {𝑥}) ⊆ 𝑏))
74	73	elrab 3331	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ↔ (𝑔 ∈ (𝑅 Cn 𝑆) ∧ (𝑔 “ {𝑥}) ⊆ 𝑏))
75	65, 71, 74	sylanbrc 695	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏})
76		inrab 3858	. . . . . . . . . . . . 13 ⊢ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = {ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)}
77		simpllr 795	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → 𝑥 ∈ ∪ 𝑅)
78	11, 12	cnf 20860	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ ∈ (𝑅 Cn 𝑆) → ℎ:∪ 𝑅⟶∪ 𝑆)
79		fdm 5964	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ:∪ 𝑅⟶∪ 𝑆 → dom ℎ = ∪ 𝑅)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ∈ (𝑅 Cn 𝑆) → dom ℎ = ∪ 𝑅)
81	80	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → dom ℎ = ∪ 𝑅)
82	77, 81	eleqtrrd 2691	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → 𝑥 ∈ dom ℎ)
83		simprr3 1104	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑎 ∩ 𝑏) = ∅)
84	83	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → (𝑎 ∩ 𝑏) = ∅)
85		sseq0 3927	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) ∧ (𝑎 ∩ 𝑏) = ∅) → (ℎ “ {𝑥}) = ∅)
86	85	expcom 450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∩ 𝑏) = ∅ → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → (ℎ “ {𝑥}) = ∅))
87	84, 86	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → (ℎ “ {𝑥}) = ∅))
88		imadisj 5403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ “ {𝑥}) = ∅ ↔ (dom ℎ ∩ {𝑥}) = ∅)
89		disjsn 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom ℎ ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ dom ℎ)
90	88, 89	bitri 263	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ “ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ dom ℎ)
91	87, 90	syl6ib 240	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → ¬ 𝑥 ∈ dom ℎ))
92	82, 91	mt2d 130	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ¬ (ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏))
93		ssin 3797	. . . . . . . . . . . . . . . 16 ⊢ (((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏) ↔ (ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏))
94	92, 93	sylnibr 318	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
95	94	ralrimiva 2949	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ∀ℎ ∈ (𝑅 Cn 𝑆) ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
96		rabeq0 3911	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)} = ∅ ↔ ∀ℎ ∈ (𝑅 Cn 𝑆) ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
97	95, 96	sylibr 223	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)} = ∅)
98	76, 97	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)
99		eleq2 2677	. . . . . . . . . . . . . 14 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → (𝑓 ∈ 𝑢 ↔ 𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎}))
100		ineq1 3769	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → (𝑢 ∩ 𝑣) = ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣))
101	100	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → ((𝑢 ∩ 𝑣) = ∅ ↔ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅))
102	99, 101	3anbi13d 1393	. . . . . . . . . . . . 13 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → ((𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ 𝑣 ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅)))
103		eleq2 2677	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → (𝑔 ∈ 𝑣 ↔ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}))
104		ineq2 3770	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}))
105	104	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → (({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅ ↔ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅))
106	103, 105	3anbi23d 1394	. . . . . . . . . . . . 13 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → ((𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ 𝑣 ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅) ↔ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)))
107	102, 106	rspc2ev 3295	. . . . . . . . . . . 12 ⊢ (({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∈ (𝑆 ^ko 𝑅) ∧ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∈ (𝑆 ^ko 𝑅) ∧ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
108	51, 53, 64, 75, 98, 107	syl113anc 1330	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
109	108	expr 641	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
110	109	rexlimdvva 3020	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
111	37, 110	syld 46	. . . . . . . 8 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
112	111	rexlimdva 3013	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (∃𝑥 ∈ ∪ 𝑅 ¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
113	25, 112	syl5bir 232	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
114	24, 113	sylbid 229	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
115	114	ex 449	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
116	9, 115	sylbird 249	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ ∪ (𝑆 ^ko 𝑅) ∧ 𝑔 ∈ ∪ (𝑆 ^ko 𝑅)) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
117	116	ralrimivv 2953	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ∀𝑓 ∈ ∪ (𝑆 ^ko 𝑅)∀𝑔 ∈ ∪ (𝑆 ^ko 𝑅)(𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
118		eqid 2610	. . 3 ⊢ ∪ (𝑆 ^ko 𝑅) = ∪ (𝑆 ^ko 𝑅)
119	118	ishaus 20936	. 2 ⊢ ((𝑆 ^ko 𝑅) ∈ Haus ↔ ((𝑆 ^ko 𝑅) ∈ Top ∧ ∀𝑓 ∈ ∪ (𝑆 ^ko 𝑅)∀𝑔 ∈ ∪ (𝑆 ^ko 𝑅)(𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ^ko 𝑅)∃𝑣 ∈ (𝑆 ^ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
120	3, 117, 119	sylanbrc 695	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ^ko 𝑅) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  dom cdm 5038   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Hauscha 20922  Compccmp 20999   ^_ko cxko 21174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-haus 20929  df-cmp 21000  df-xko 21176

This theorem is referenced by: (None)